Since it sounds like there is no talking you out of keeping a PBT in a 125 gallon tank, here is my humble opinion and experience with a PBT in that size tank. A standard 180 and a standard 125 are both typically 6' in length, so it really doesn't matter all that much in regards to which tank it "will do better in". In the long term, a 6' tank is not enough swim room for most Acanthurus once they hit the 4-6" range. Acanthurus are known to become belligerant and downright aggressive and stressed when they feel cramped. I kept a gorgous PBT in my current 180 until it reach the 4.5" mark, then it had to be removed and sold to a friend with a much larger tank. It was aggressive to anything that moved, including my hand and arms. I have also kept an Achille's tang in an 8' 300 gallon display and it became a complete bully and would attack larger Angels.
So basically the question is "Can you keep a PBT in a 125?", and technically the answer is "Yes". However do you want to keep the PBT healthy and alleviate aggression in the long term, then the answer is look for a larger tank or skip the PBT.There are plenty of people on these forums that will argue the contrary.
BTW, I sucessfully used tank transfer and Hypo to treat both specimens. I also run Ozone, large UVs, and an oversized skimmer which I believe helped keep the PBT and Achille's free of parasites. Good Luck!